Cultural Dynamics, Social Mobility and Urban

Cultural Dynamics, Social Mobility and Urban Segregation∗
Emeline BEZIN†and Fabien MOIZEAU‡
January 2017
Revised version
Abstract
We consider the relationship between cultural dynamics, urban segregation and inequality.
To this end, we develop a model of neighbourhood formation and cultural transmission. The
tension between culture preservation and socioeconomic integration drives the pattern of segregation in the city. We study the dynamics of culture and urban configurations. In the long
run, the city may end-up segregated or integrated depending on cultural distance and the initial
cultural composition of the population. We also show that segregation fosters the influence of
family background on economic fate. Finally, segregation has ambiguous effects for long-run
efficiency.
Keywords: cultural transmission, peer effects, residential segregation, human capital inequality.
JEL Classification: D31, I24, R23.
∗
We thank the Editor, Gilles Duranton, and two anonymous referees for very thoughtful and detailed comments. We
are very grateful to David de la Croix, Luisa Gagliardi, Victoire Girard, François Salanié and Thierry Verdier for their
helpful comments. We thank participants at the summer school “Social Interactions and Urban Segregation” (Rennes,
2014), the 9th Meeting of the Urban Economics Association (Washington, 2014), the 64th AFSE congress (Rennes,
2015), the 30th Annual congress of the EEA (Mannheim, 2015), the 12th annual conference of TEPP (Paris), and
seminar participants at ENS Cachan, BETA (Strasbourg), EconomiX (Nanterre), Université Saint-Louis (Bruxelles),
the CREM-SMART workshop (Rennes), PUCA (Urban Development Construction and Architecture, 2015) and IDEJETRO (Chiba). Financial support from the Agence Nationale de la Recherche (ANR-12-INEG-0002) is gratefully
acknowledged.
†
Paris School of Economics (PSE), E-mail: [email protected]
‡
CREM (Condorcet Center), Université de Rennes 1 and Institut Universitaire de France, E-mail:
[email protected].
1
1
Introduction
During recent decades, most Western democracies have become more ethnically and culturally diverse. The average proportion of foreign-born individuals in OECD countries rose from 9.5% in 2000
to 13% in 2014 (OECD International Migration Outlook, 2016). This movement is likely to continue,
given demographic and migration trends. Increasing diversity challenges social cohesion and puts
issues of social integration and national identity at the forefront of the political debate.
In the host country, ethnic minorities often live in the less affluent neighborhoods of metropolitan
areas. Living in ethnic enclaves produces both benefits and costs for inhabitants suggesting that the
choice of place of residence results from a variety of incentives. It responds to the desire to live close
to the native population in order to acquire the mainstream culture and become socially-integrated,
but also both the wish to cluster with peers and retain the cultural attitudes of the country of origin,
sometimes at the expense of social integration.
Understanding how residential segregation affects the incentives to socially integrate and preserve
the home-country culture is essential for the understanding of potential policies to reduce the ethnic
gap.
This paper analyzes the interdependency between cultural transmission, urban segregation and
economic inequality. Our framework allows us to consider, on the one hand, how segregation influences the way in which cultural traits are passed on from one generation to the next and, on the other
hand, how cultural transmission drives the incentives to segregate. We are thus able to answer the
following questions: How does segregation contribute to cultural diversity within the society? How
does the existence of diverse cultures regarding personal achievement affect segregation and urban
inequality? How can we design public policies to affect both segregation and cultural transmission
in order to improve societal economic performance?
It is well-documented that urban segregation interacts with culture (regarded as preferences,
beliefs and social norms). Urban segregation influences ethnic identity, although there is no consensus
on the sign of this relationship (see Bisin, Patacchini, Verdier and Zenou, 2011a, and Constant,
Schüller and Zimmerman, 2013). Segregation of ethnic minorities in poor neighborhoods creates a
‘culture of poverty’ by socially isolating individuals from mainstream norms of behavior (see Wilson,
1987, Anderson, 1999, Lamont and Small, 2008). The choice of the social arenas in which children
interact such as schools or neighborhoods is also a concern for parents who care about the transmission
of desired cultural traits (see, for the particular case of school choice, Ioannides and Zanella, 2008,
2
or Tinker and Smart, 2012).
Following on from this empirical evidence, we develop a theoretical model based on the following
three blocks.
First, the population consists of two different cultural types: the culture of the majority (say
that of natives) and the minority culture (say that of the foreign-born). We assume that agents who
adopt different cultures do not have the same prospects of economic success (i.e. being educated),
with the majority culture performing better as it produces better knowledge of the codes of behavior
and the functionings of the schooling system.
Second, cultural traits are transmitted intergenerationally following a process à la Bisin and
Verdier (2001). Interactions within the family and within society are involved here, and parents have
an incentive to socialize children into their own culture.
Third, parents choose the place where they wish to live. This choice is not only motivated by
the desire to transmit one own’s culture but also by the existence of local peer effects in children’s
education (see, for instance, Bénabou, 1993, 1996a, b). Local spillovers matter as, whatever their
cultural trait, all parents value having educated children.
To capture the influence of culture on socioeconomic outcomes, one crucial feature of our model is
that the (subjective) benefits of education and the gain associated with the transmission of cultural
traits are linked. More precisely, for mainstream parents, we consider that the benefit of having
an educated child rises when the child has acquired the parents’ own culture (i.e. the mainstream
culture). The mainstream cultural trait and education are thus complements. For minority families,
we consider two cases: complementarity or substitutability. Under substitutability, having the minority cultural trait reduces the benefits of education. These two cases capture the cultural distance
between mainstream and minority groups. Complementarity for both groups reflects cultural proximity. While substitutability for the minority group corresponds to cultural polarization. There is
empirical support that cultural distance matters for differences in socioeconomic outcomes (for the
impact of religion on economic decisions, see Weber, 1958, or Botticini and Eckstein, 2005, 2007, for
the influence of cultural origin on social integration of immigrants, see Domingues Dos Santos and
Wolff, 2011, for french evidence, or Gang and Zimmerman, 2000, for german evidence, and Borjas,
1995, for US evidence, for the influence of oppositional identities see Akerlof and Kranton, 2002,
Fryer and Torelli, 2010, Battu and Zenou, 2010, and Battu et al. 2007).
The non-separability between the benefit of education and the gain associated with culturaltrait transmission means that the incentives parents face to transmit cultural trait and to make
3
their offspring educated are intertwined and influence the integration and segregation forces. The
main insight of our theory is then that the urban equilibrium and the cultural composition of the
population are co-determined. We show that cultural distance has crucial implications for the nature
of the long-run equilibrium.
When there is cultural polarization, the desires to preserve the minority culture and socially
integrate are contradictory, making minority parents less willing to pay to live in better-quality
neighborhoods. The segregation force is then strong enough so that the city ends-up segregated.
When there is cultural proximity, there are multiple types of long-run urban configurations. We
show that the long run urban equilibrium depends on society’s initial cultural composition.
We show that the spatial separation of cultural groups adds further glue to the intergenerational
transmission of cultural traits. Consistent with the findings in Borjas (1995) and Chetty et al. (2014),
segregation thus strengthens the influence of family background on economic fate.
We show that segregation has ambiguous effects on the long-run level of education. The initial
population cultural composition is key to assess the efficiency of the urban equilibrium.
Related literature. Our paper is related to the literature on cultural transmission launched by
Bisin and Verdier (2001). The transmission of the traits such as identities, time preferences and
beliefs, which impact educational outcomes, has been analyzed theoretically (see Bisin, Patacchini,
Verdier, and Zenou, 2011b, for oppositional identities, Doepke and Zilliboti, 2008, for time preferences
and the spirit of capitalism, Guiso, Sapienza and Zingales, 2008, for beliefs and trust in other people,
and Lindbeck and Nyberg, 2006, for the transmission of working norms). Our paper is relatively
close to some theoretical and empirical studies suggesting that assimilation policies can lead to
a cultural backlash from the minority (Bisin, Patacchini, Verdier, and Zenou, 2011b, Carvalho,
2013, and Fouka, 2016). In the same vein, Verdier and Zenou (2017) shows how cultural distance
(defined as the degree of centrality in a network) affects choices of assimilation. None of the previous
studies consider location choices and is able to show how cultural choices interact with the degree of
segregation.
Our paper also contributes to the literature on neighborhood effects and endogenous socioeconomic segregation explaining how local interactions drive spatial segregation and persistent income
inequality (see for instance, Loury, 1977, Bénabou, 1993, 1996a,b, Borjas, 1998, and Durlauf, 1996).
In these analyses, the dynamics of income inequality rely on human-capital accumulation, and individual human capital is determined by both that of their parents and local spillovers. In particular,
4
Bénabou’s works emphasize that incentives to segregate into distinct communities are driven by the
desire to enhance human-capital accumulation. Departing from Bénabou’s works, Borjas (1998) introduces ethnic spillovers in the human-capital accumulation process that lead ethnic groups to sort
across neighborhoods. In the same vein, we consider that cultural aspects are crucial for the emergence of the urban configuration. Moizeau (2015) also studies the influence of culture on residential
choices. His analysis considers how in a city either opposing social norms persist or a particular
code of behavior spreads and ultimately prevails. The dynamics of cultural traits follow a particular
diffusion process proposed by Akerlof (1980). We differ from these previous works as the cultural
composition of the population evolves over time as a result of individual decisions. Our approach
allows us to take into account the tension between the desire to preserve one’s own culture and the
need to integrate in order to improve one’s prospect of economic success. To the best of our knowledge, our paper is the first to emphasize how this tension between culture and economic integration
impacts cultural diversity and residential segregation in the long run.
Our paper is also related to Card, Mas and Rothstein (2007, 2008), who build a model à la
Schelling where individuals have preferences over the social environment. Unlike most theoretical
models of neighborhood composition, they find that tipping dynamics may lead to multiple long-run
equilibria, with integration being a stable outcome. Our cultural explanation of multiple long-run
urban configurations here relates the degree of segregation to cultural distance, as well as the cultural
composition of the population. It is thus consistent with the empirical findings in Cutler, Glaeser and
Vigdor (2008) that (i) the cultural distance between an immigrant group and the native population
significantly affects the degree of segregation, and (ii) the group share in the population also matters
for the urban configuration.
The remainder of our paper is organized as follows. The following section sets out the model.
Section 3 then provides a characterization of the segregation that emerges at each date t, and looks
at the dynamics of urban segregation and cultural traits. In Section 4, we present the result that
both integration and segregation can be long-run outcomes, and Section 5 addresses the issue of
efficiency in the urban equilibrium. Last, Section 6 concludes.
5
2
2.1
The set-up
The city
The city is comprised of two residential areas indexed by j = 1, 2. We consider that the land rent is
paid to absentee landlords, and without loss of generality normalize the opportunity cost of building
a house to 0. Houses are identical across the city. The inelastic supply of houses within a residential
area is of mass 1. This land-market is a closed-city model where the population of the city is a
continuum of families of mass 2. Each family, comprised of a parent and a child, lives in one and
only one house. The city can accommodate the entire population. Agents live two periods. As a
child , the individual is subject to socialization and attends school. As an adult, the individual has
to decide in which neighborhood her family will live, and the effort to exert to transmit her cultural
trait.
2.2
Cultural transmission, peer effects and preferences
Parents differ with respect to their cultural trait. They have either trait a which refers to the
mainstream culture group or trait b which refers to the minority culture group. At any date t, the
city population is comprised of Qt type a parents and 2 − Qt type b parents. We denote by qtj , the
number, resp. fraction, of agents with trait a in area j.
Cultural transmission. The transmission of preferences follows the lines of the model introduced
by Bisin and Verdier (2001). The intergenerational transmission of trait i ∈ {a, b} is the result of
social interactions which arise at two levels. The child born at date t is first exposed to vertical
socialization by her/his parents. The probability that the latter directly transmits her/his trait is
τ i . If not socialized within the family (with probability 1 − τ i ), the child adopts the trait of some
role model met in neighborhood j. This second socialization process is called oblique transmission.
The probability of being obliquely socialized into trait a (resp. b) in neighborhood j is f (qj ) (resp.
1 − f (qj )). We assume that oblique transmission of trait a increases with the fraction of role models
with trait a, i.e. f 0 (qj ) > 0, and that f (0) = 0 and f (1) = 1. As stated by Saez-Marti and Sjögren
(2008) and Saez-Marti and Zenou (2012), the shape of f (.) captures the degree of conformism, that
is how much does the child finds attractive the trait a acquired by role models. When f 0 (0) < 1 and
f 0 (1) < 1 (resp. f 0 (0) > 1 and f 0 (1) > 1), the child is more inclined to imitate (resp. distinguish
from) role models who are more frequent.
6
The transition probabilities are given by
ab
aa
= (1 − τ a )(1 − f (qj ))
= τ a + (1 − τ a )f (qj ) and Pj,t
Pj,t
(1)
bb
ba
Pj,t
= τ b + (1 − τ b )(1 − f (qj )) and Pj,t
= (1 − τ b )f (qj ).
(2)
aa
denotes the probability that a child from a type a family be socialized into type a
In particular, Pj,t
at time t.
Peer effects. The young generation attends school. We do not consider any educational effort but
we allow the probability to get education to differ across cultural type i. Further, the probability
ē|i
e|i
ē|i
of educational success depends on neighborhood j. Denoting by Pj,t (resp. Pj,t = 1 − Pj,t ) the
probability of getting education (resp. no education), we assume:
ē|i
Assumption 1 Suppose that Pj,t is a function of qj,t and for any qj,t and j = 1, 2
ē|a
ē|b
(i) Pj,t > Pj,t ,
ē|a
ē|b
dPj,t
dPj,t
(ii)
≥
> 0.
dqj,t
dqj,t
Children do not face the same educational opportunities whatever their type due to cognitive
and non-cognitives abilities. Further, trait a is more favorable to education than trait b so that the
fraction of mainstream agents who get education is higher than the fraction of minority agents who do
so. Several reasons can explain this educational gap. The minority population lacks the knowledge of
the social codes and the characteristics of the school system such as the quality of teaching program,
the contents and type of educational curricula. The fact that the minority population may have to
cope with a foreign language also contributes to this gap (see Dustmann, Frattini and Lanzara, 2012).
Our assumption is also consistent with Borjas (1995) result that ethnicity continues to be correlated
with children’s human capital once parental skills and average human capital in the neighborhood
have been controlled for.
Also we assume peer effects in education: the higher the fraction of type a individuals, the higher
the fraction of people who educate in the population (due to (i)) and the higher the incentives to
educate whatever one’s cultural trait1 .
1
There is an extensive literature on the impact of neighborhood effects on individual socioeconomic outcomes (see,
for instance, surveys of Durlauf, 2004, and Topa and Zenou, 2014). We should mention here that there is not yet a
consensus about the size of neighborhood effects on educational outcomes. Some experimental or quasi-experimental
7
Preferences. Parental preferences depend on private consumption and an altruistic component
which depends on both offspring’s educational attainment and cultural type. Specifically, V ē|ii
0
denotes the gain from the type i parent’s point of view that a child gets educational level e and
0
acquires trait i0 . V e|ii is this gain when the child gets educational level e which is lower than its
0
counterpart e. For the sake of simplicity, V e|ii is exogenous. W
We denote by Uti (ρj,t , τ i ), the utility at date t of a parent with trait i and income w, who lives in
neighborhood j and exerts socialization effort τ i . We have for a trait-a parent
ē|a
e|a
ē|b
e|b
aa
ab
Uta (ρj,t , τ a ) = w − ρj,t + Pj,t
(Pj,t V ē|aa + Pj,t V e|aa ) + Pj,t
(Pj,t V ē|ab + Pj,t V e|ab ) − Θ(τ a ),
(3)
where Θ(.) is the socialization cost which is increasing and convex with respect to τ .2
A parent with trait b and income w who lives in neighborhood j ∈ {1, 2} has utility
ē|b
e|b
ē|a
e|a
ba
bb
Utb (ρj,t , τ b ) = w − ρj,t + Pj,t
(Pj,t V ē|ba + Pj,t V e|ba ) − Θ(τ b )
(Pj,t V ē|bb + Pj,t V e|bb ) + Pj,t
(4)
We make the following assumption.
Assumption 2 r
(i) Cultural intolerance: V e|ii > V e|ii
0
0
0
∀e ∈ {e, e}, ∀i, i0 ∈ {a, b} with i0 6= i,
0
(ii) Educational gain: ∆V ii ≡ V ē|ii − V e|ii > 0
∀i, i0 ∈ {a, b},
(iii) Complementarity between education and mainstream cultural trait: ∆V aa > ∆V ab .
Following the literature on cultural transmission (Bisin and Verdier, 2001), item (i) assumes that
preferences embody cultural intolerance: for a given educational level, a parent prefers a child with
her own cultural trait. Item (ii) amounts to say that whatever her trait, the parents prefers that her
child gets education. This assumption captures the idea that, eventhough educational gain measured
0
by the magnitude of ∆V ii differs across cultural traits, there is a widespread view that education
pays. Item (iii) considers complementarity between education and cultural trait a meaning that, for
trait-a parents, educational gain is magnified when the child acquires trait a. We will say that the
work finds little evidence of neighbourhood effects on educational outcomes (see Kling et al. 2007, Oreopoulos,
2003). Topa and Zenou (2014) provide an interesting discussion of why these experimental analyses may lead to
insignificant treatment effects on economic outcomes. A growing literature in sociology emphasises the duration of
exposure to neighbourhoods, which helps explain why experimental work finds little evidence of neighbourhood effects
(see Sharkey and Elwert, 2011, Wodtke et al. 2011). Chetty, Hendren and Katz (2015) consider the Moving To
Opportunity Experiment and show that treatment effects are substantial when considering the duration of exposure
to a better neighbourhood.
2
In order to focus on the impact of cultural traits and rule out any income heterogeneity effect, we assume a linear
utility function of private consumption w − ρj,t .
8
mainstream culture values education. We make no assumption regarding trait b and we will examine
alternatively two cases, that is complementarity (∆V bb > ∆V ba ) and substitutability between trait
and education (∆V bb < ∆V ba ). Both cases capture cultural distance, we will say that:
Definition 1 (i) Cultural proximity refers to complementarity between culture and education for both
mainstream and minority, (ii) cultural polarization refers to complementarity for the mainstream and
substitutability for the minority.
This non-separability assumption is a key feature of our framework and implies that incentives
to exert socialization effort and incentives to make the offspring get education are intertwined.
The crucial feature is that parents have two concerns regarding the welfare of their child. First,
given cultural intolerance, parents value to have a child with the same cultural trait. Second, they
also prefer to have an educated child. Both concerns may be compatible when cultural trait and
education are complements while coming into contradiction under substitutability. Hence, when
choosing socialization effort and location, parents face a trade-off between cultural transmission and
education that hinges on the degree of substitutability between culture and education and also on
the endogenous composition of the population.
2.3
Parental choices: socialization and location
Parents make two decisions. They choose both the location j where they pay the land rent ρj,t and
their socialization effort.
Socialization choice. Direct transmission is the result of a choice. Parents exert the effort τ i ∈
[0, 1] in order to transmit their trait. Let us first consider trait a parents. At date t, given her place
of residence j, a type a parent chooses her optimal effort τ ∗a that solves
max
Uta (ρj,t , τ a ) subject to (1).
a
τ
For the sake of presentation, we omit the time index t and the neighborhood index j when not
0
necessary. Given that P ē|i = 1 − P e|i , and P ii = 1 − P ii for any i, i0 ∈ {a, b}, i 6= i0 , we get the
following first-order condition:
(1 − f (q)) P ē|a ∆V aa − P ē|b ∆V ab + V e|aa − V e|ab = Θ0 (τ ∗a ).
9
(5)
We assume that for any q
P ē|a ∆V aa − P ē|b ∆V ab + V e|aa − V e|ab ≥ 0
(6)
implying that there is always an incentive to exert a socialization effort, i.e. τ ∗a (q) ≥ 0.3
From (5), applying the implicit function theorem leads to:
ē|a
−f 0 (q)(P ē|a ∆V aa − P ē|b ∆V ab + V e|aa − V e|ab ) + (1 − f (q)) dPdq
dτ ∗a
=
dq
Θ00 (τ ∗a )
∆V aa − ∆V ab
.
(7)
The above expression shows that q has two effects on the socialization effort. First, when a parent is
surrounded by more people with the same cultural trait, she benefits from a more effective oblique
transmission and is incited to reduce her socialization effort. This negative effect, captured by the
first parenthesis in the numerator of (7), is called the cultural substitution property (see Bisin and
Verdier, 2001). Our model involves a second effect: when the fraction of type a agents rises, peer
effects in education are stronger and the child is more likely to educate. As cultural trait a and
education are complements, this positively affects the incentives to socialize children. This second
effect may counteract the cultural substitution effect. Hence, the impact of q on τ ∗a is ambiguous.
Further, when the neighborhood population is only comprised of trait a individuals, the cultural
substitution effect prevails as τ ∗a (1) = 0.
Following the same reasoning, we obtain the socialization effort of trait b parents (see Appendix
7.1). The socialization choice of minority parents depends on a cultural substitution effect. Parents
with trait b have less incentive to socialize their child when there are more type b individuals in
the neighborhood, i.e. when q is lower. The socialization choice also depends on educational gains
weighted by peer effects. An increase in peer effects (captured by a rise of q) has a positive (resp.
negative), impact on socialization effort if cultural trait b and education are complements (resp.
substitutes). Hence, the overall impact of q on τ ∗b has an ambiguous sign under substitutability
while it is positive under complementarity. Further, when the neighborhood is inhabited by only
type b individuals, only cultural substitution matters yielding τ ∗b (0) = 0.
Vertical and oblique transmissions drive the dynamics of the mainstream culture in neighborhood
We could relax this assumption and have values of q such τ ∗a (q) = 0. This would not add any interesting insights
to our discussion.
3
10
j between t and t + 1
qt+1 − qt = (1 − τ ∗b (qt ))f (qt ) − qt (1 − τ ∗a (qt )) + qt f (qt )(τ ∗b (qt ) − τ ∗a (qt )).
Parents’ Location Choice.
(8)
In this framework, the location choice by determining the socioeco-
nomic composition of the neighborhood is another mean used by parents to influence both cultural
transmission and educational outcome of their child.
At any date t, the location choice of parents with trait i solves the following program
max Uti (ρj,t , τ ∗i (qj,t )).
j
Without loss of generality, we impose that q1,t ≥ q2,t = Qt − q1,t and ρ2,t = 0. Following the
literature, the urban equilibrium is defined as follows:
∗
∗
),
, τ ∗a (q1,t
Definition 2 At any date t, given Qt , the urban configuration characterized by ρ∗t , q1,t
∗
∗
∗
τ ∗a (Qt − q1,t
), τ ∗b (q1,t
), τ ∗b (Qt − q1,t
) is an equilibrium if no one wants to move and change their
socialization choice.
The urban equilibrium is spatially stable if, after a move of a small number of trait-a individuals
from neighborhood 1 to neighborhood 2 and a migration of the same number of trait-b individuals in
the reverse direction, the highest bidders for neighborhood 1 are trait-a individuals.
To obtain the urban equilibria, we need to know who is eager to bid more for land in a particular
neighborhood. Differing from the monocentric city benchmark (Duranton and Puga, 2015), the bid
curve refers to the willingness to live in urban area 1 as a function of the share of the mainstream
group. Specifically, the willingness to pay to live in urban area 1, denoted by ρi for i ∈ {a, b}, is such
that a trait-i parent is indifferent between both neighborhoods. For a mainstream individual, given
ē|a
that Pjab = 1 − Pjaa and Pj
ē|a
e|a
= 1 − Pj , we have
ē|b
ē|a
ē|b
ρa = P1aa (P1 ∆V aa − P1 ∆V ab + V e|aa − V e|ab ) − P2aa (P2 ∆V aa − P2 ∆V ab + V e|aa − V e|ab )
ē|b
ē|b
+(P1 − P2 )∆V ab − (Θ(τ ∗a (q1 )) − Θ(τ ∗a (q2 ))).
11
(9)
ē|b
e|b
For a minority individual, using Pjba = 1 − Pjbb and Pj = 1 − Pj , we obtain
ē|b
ē|a
ē|b
ē|a
ρb = P1bb (P1 ∆V bb − P1 ∆V ba + V e|bb − V e|ba ) − P2bb (P2 ∆V bb − P2 ∆V ba + V e|bb − V e|ba )
ē|a
ē|a
+(P1 − P2 )∆V ba − (Θ(τ ∗b (q1 )) − Θ(τ ∗b (q2 ))).
(10)
The equilibrium in the land market is segregated (resp. integrated) when trait-a parents are more
(resp. less) willing to pay to live in urban area 1 than trait-b parents. Formally, the urban equilibrium
can be derived from the ranking of the slopes of the bid-rent curves (available in Appendix 7.2). We
thus highlight the forces that lead to either segregation or integration. They are triggered by the
desire to transmit the cultural trait and the concern to have an educated child. First, cultural
intolerance is a force leading to segregation. Because of transmission by role models, the probability
to have a child with the same trait is higher when many people share this cultural trait in the
neighborhood. Cultural intolerance creates an incentive for each type of agents to cluster together
in part. Second, given the concern to have an educated child, parents search for peer effects in
education, thus generating an incentive to live in the neighborhood where the mainstream cultural
trait prevails. How this motive affects the incentives to segregate depends on cultural distance.
3
Segregated urban equilibria
We first study the conditions that allow segregation to arise at each date t. We make a symmetry
assumption ∆V aa = ∆V ba and ∆V ab = ∆V bb in order to focus on the consequences of cultural
distance in terms of segregation (for an asymmetric case see Proof of Proposition 1 in the Appendix).
Proposition 1 If there is cultural polarization then the unique stable urban equilibrium is always
segregated: if Q < 1 then q1∗ = Q and q2∗ = 0, if Q ≥ 1, q1∗ = 1 and q2∗ = Q − 1.
Cultural polarization leads to segregation. The reason is that cultural transmission of the minority trait and education are conflicting objectives implying that when they live in neighborhood 1
(searching for peer effects), parents with the minority trait gain more from having a child with the
mainstream cultural trait. However, the probability to have a child with the mainstream cultural
trait is always lower for minority parents than for mainstream ones (i.e. P1aa > P1ba ) as the latter positively impact this probability by exerting a direct socialization effort. This reduces the gain to live
in neighborhood 1 for parents b compared to parents a. Hence, when there is cultural polarization,
12
mainstream parents are willing to bid more than minority ones and the city is segregated4 .
ē|a
ē|b
Remark that in the absence of peer effects, i.e. dPj /dqj =dPj /dqj = 0, only cultural intolerance matters leading the urban equilibrium to be segregated. This highlights a new segregation
force which results from the desire to preserve one’s own culture differing from the ones studied in
the literature on neighborhood effects (see Bénabou, 1993, 1996a,b or de Bartolome, 1990).
The segregated equilibrium exhibits spatial disparities in terms of cultural transmission and educational rate. This comes from the fact that chances to acquire the mainstream cultural trait for
children living in urban area 1 are better than for those living in urban area 2. Hence, it can be
checked that the probability that a child of trait-a parents be socialized into this trait is always higher
in urban area 1. Furthermore, chances to acquire trait a and have greater probability to educate for
children of trait-b parents are better in urban area 1. This difference translates into higher education
in urban area 1. When Q < 1, urban area 1 is characterized by a positive rate of education while
urban area 2 is populated only by minority group reaching a minimal rate of education. When Q > 1,
urban area 1 is populated by the mainstream group reaching a maximal rate of education while the
mixed urban area 2 is characterized by a lower fraction of youth population who gets education.
Further, our framework allows to get insights on the influence of segregation on persistence of the
cultural trait across generations. We measure intergenerational cultural mobility by the gap between
probabilities to get trait a conditional on the family trait:
P aa − P ba =
1 1
[q1 P1aa + q2 P2aa ] −
(1 − q1 )P1ba + (1 − q2 )P2ba .
Q
2−Q
(11)
with transition probabilities depending on equilibrium socialization efforts.
In order to have a tractable result of the impact of segregation on cultural mobility, we work with
the following specific forms P ē|i = αi q, with αa > αb , f (q) = q and Θ(τ ) = (1/2θ)τ 2 and show that
Corollary 1 When the cost of socialization is high, cultural family background is more persistent
across generations under segregation than under integration.
As in standard models of cultural transmission, our framework gives rise to persistence of cultural
traits across generations, i.e. P aa − P ba > 0. This is due to the existence of a bias in the transmission
process induced by the socialization effort of parents. What is more is that segregation adds further
4
When a segregated equilibrium exists, it is always spatially stable. Indeed, there is always a trait i population
which strictly prefers her place of residence, i.e. trait a population when Q < 1 and trait b population otherwise.
A small perturbation of the segregated equilibrium does not change the identity of the highest bidder for land in
neighborhood 1.
13
glue in the cultural transmission process. The intuition goes as follows. Segregation (as compared
to integration) has two distinct effects on the probability differential, P aa − P ba . First, segregation
increases oblique transmission of trait a for both types of child. However, this effect is stronger for
children from mainstream family background. Hence, the oblique transmission effect favours the rise
of the probability differential P aa −P ba . Second, a rise in q affects socialization efforts. More precisely,
due to cultural substitution, an increase in q decreases the effort of parents a which negatively affects
the probability differential. When the cost of socialization is high, the cultural substitution effect
is low. Hence, the oblique transmission effect prevails and the rise in segregation positively impacts
the probability differential.
Given Assumption 1, cultural mobility is positively related to social mobility. Corollary 1 is
supported by empirical evidence. Borjas (1995) finds that the rate of mean convergence of skills
between ethnic groups is reduced when people live in segregated neighborhoods and are more exposed
to ethnicity influence. Recent work on intergenerational mobility in the US has emphasized the role
of neighborhoods for individual prospects of social mobility (see Chetty, Hendren, Kline and Saez,
2014, Chetty and Hendren, 2015). In particular, Chetty, et al. (2014) use data from Federal income
tax records over the 1996-2012 period to show the striking spatial variation in social mobility. They
also show that high-mobility areas have less residential segregation and income inequality, better
primary schools, and greater social capital and family stability.
We now turn to the cultural dynamics that emerge when the city is segregated.
Assumption 3 Suppose that
(i) τ a (0) < 1 − f 0 (0),
(ii) τ b (1) > 1 − f 0 (1),
(iii) ∃ q ∈]0, 1[ such that f (q) ≥ q and τ a (q) ≥ τ b (q).
Assumption 3 clarifies the strength of the different forces at stake in the dynamics of cultural
traits. According to (8), the dynamics of cultural traits is driven by three distinct forces: (i) the
degree of conformism captured by the shape of f , (ii) peer effects in education, and (iii) cultural
substitution where the two last forces are embodied in the socialization efforts τ a and τ b .
High conformism and peer effects favor homogeneity within the population. Reversely, cultural
14
substitution is a force leading to heterogeneity. Item (i) makes statement about these forces when
the mainstream population is low (i.e. q = 0). Both peer effects and conformism are then sufficiently
high to overcome the cultural substitution effect (meaning that the socialization effort of mainstream
agents is below some threshold). Item (ii) focuses on large fractions of mainstream agents (i.e. q = 1).
Conformism and peer effects are then not sufficiently high to overcome cultural substitution (implying
that the socialization effort of minority agents is above some threshold). Item (iii) imposes that for
some intermediate value of the population composition, conformism and peer effects are high.
Although empirical evidence on socialization rates and cultural transmission is still sparse, we
can find some support for Assumption 3 in Bisin, Topa and Verdier (2004). From a structural
estimation of a cultural transmission model, they find that socialization efforts of some religious
group is positively related to the share of this group (what they call cultural complementarity). In
other words, socialization rates are lower the smaller are religious shares, thus justifying item (i).
Nonetheless, they find that socialization choices of minorities (Jews) is much higher than dominant
groups (Protestants and Catholics) when the minority share is close to 0 giving support for item (ii).
By Assumption 3, we consider a case where the different forces at stake matter for the dynamics
of culture. Since this framework generates homogeneity forces as well as heterogeneity forces, this
allows for a general situation where, in the long run, a culturally diverse as well as a culturally
homogenous society can emerge.
Proposition 2 Under Assumption 3, the population dynamics in the segregated city exhibits at least
two stable stationary equilibria: Q = 0 and Q = q ∗ < 1.
In the long run, population dynamics converge to either cultural homogeneity or cultural diversity. This is due to the interplay between opposite forces that drive cultural transmission. With
Assumption 3, we consider a general case where both types of forces can prevail depending on the
composition of the population.
Remarkably, because we assume that the cultural substitution effect is high, there exists a longrun segregated equilibrium (Q = q ∗ ) such that neighborhood 1 is culturally diverse. By contrast,
in neighborhood 2 with only the minority culture, given item (i) of Assumption 3, the dynamics of
cultural trait push toward homogeneity provided.
We can show that the persistence of cultural diversity in one neighborhood has important implications for long-run economic inequalities. Let us define inequalities by the educational gap between
15
a mainstream individual and its counterpart, i.e. P e|a − P e|b . It is expressed as follows
e|a
P
e|a
−P
e|b
e|a
q 1 P 1 + q 2 P1
=
Q
e|b
e|b
(1 − q1 )P1 + (1 − q2 )P2
−
.
(2 − Q)
(12)
At the long-run equilibrium Q = q ∗ < 1, the educational gap equals
e|b
e|a
P1 (q ∗ )
1 − q ∗ e|b ∗
P (0)
−
P1 (q ) − 2 ∗ .
∗
2−q
2−q
(13)
Let us compare this gap with the one which would be achieved at a long-run equilibrium where both
neighborhoods are culturally homogeneous, i.e. q ∗ = 1 which would arise if we would assume that
cultural substitution is low5 . In the long run, the educational gap is
e|a
e|b
P1 (1) − P1 (0).
e|b
(14)
e|b
From (13) and (14) and as P1 (0) = P2 (0) , the long-run equilibrium with Q = q ∗ < 1 leads to
lower inequality than the one with q ∗ = 1 if and only if
1 − q ∗ e|b
e|b
e|a
e|a
(P1 (0) − P1 (q ∗ )) < P1 (1) − P1 (q ∗ ),
∗
2−q
which is true given the existence of peer effects. From the inequality perspective, the long-run
stationary equilibrium where one neighborhood is culturally diverse is thus more desirable than the
long-run equilibrium with homogeneous neighborhoods. Hence, our framework emphasizes that the
cultural substitution property which leads to cultural diversity is crucial for the long-run level of
inequalities.
However, it is also likely that segregation pushes to cultural homogeneity in the city. This case
occurs when in neighborhood 1 where both cultural groups live cultural substitution is not strong
enough. Let us remark that Proposition 2 does not precisely characterize the basin of attraction
of these two equilibria because it requires specification of the oblique transmission function f . We
could set conditions on f , allowing to avoid cycles and then say that for low (resp. high) values of
q0 the city converges to Q = 0 (resp. Q = q ∗ ).
5
More formally, this amounts to assume that item (ii) in Assumption 3 does not hold anymore, i.e. for high
fractions of type a individuals, conformism outweighs the effect of cultural substitution i.e. τ b (1) < 1 − f 0 (1).
16
4
Integration and multiple long run cities
Our framework also allows integration to arise. As in Section 3, to make clearer where the main
results come from, we impose symmetry, that is ∆V aa = ∆V bb and ∆V ab = ∆V ba .
Proposition 3 At each date t, the integrated equilibrium q1∗ = q2∗ = Q/2 exists and is spatially stable
only if:
(i) there is cultural proximity,
(ii) transition probabilities are such that P1aa < P1bb .
This Proposition highlights that an integrated city can emerge, at each date t, provided that
the minority culture values education. A necessary condition for the integrated equilibrium to be
stable is that cultural trait and education are complements and that the probability to transmit one’s
cultural trait is relatively high for minority individuals (as compared to mainstream).
Compared to the case of cultural polarization, when there is cultural proximity, benefits from
education are high for minority parents as having an educated child does not necessarily implies a
cultural loss. Since the benefit of education increases when the child acquires the parental trait,
parents who benefit more from higher peer effects are those who are more likely to transmit their
own trait in neighborhood 1. This is the case for minority parents when P1aa < P1bb (note that this
generally arises when parents of type b form a large share of the population).6
Both Propositions 1 and 3 deliver the message that cultural distance matters for the type of
urban equilibrium. This echoes empirical findings of Cutler, Glaeser and Vigdor (2008) that cultural
distance (as measured either by linguistic differences, regional development gap or geographical
proximity) accounts for a substantial part of the degree of segregation (for instance the changing
country-of-origin composition of the immigrant population explains virtually all of the post-1970
increases in segregation).
As at the integrated equilibrium, both neighborhoods have the same cultural composition, the
education rate as well the cultural transmission process do not vary spatially. Still, there is persistence of the cultural trait within dynasties due to the fact that a cultural group has the possibility
As it can be easily checked from bid rents (9) and (10), an integrated equilibrium with q1∗ = q2∗ always exists.
However, it is not always spatially stable. Under complementarity, parents with trait b prefer to have educated
children with their own cultural trait. After a perturbation of the integrated equilibrium that marginally increases
q1∗ , neighborhood 1 provides a better educational environment with higher peer effects. Trait-b parents are able to
bid for land higher than trait-a parents only if they value more than their counterparts educational gains obtained in
neighborhood 1. This possibility arises when the probability to have a child of their own type is higher for parents
b given that both cultural traits benefit from similar peer effects and that the surplus of educational gains obtained
when the child acquires the parental cultural trait is the same among cultural traits (as we assume symmetry).
6
17
to transmit her own trait by vertical as well as horizontal transmission. Contrary to segregation,
integration does not generate any spatial bias in the oblique transmission process.
According to both items (i) and (ii) of Proposition 3, segregation as well as integration can emerge
in the short run. The following proposition provides a characterization of the urban configurations
arising in the long run.
Proposition 4 Suppose that there is cultural proximity, the integrated equilibrium with (q1∗ , q2∗ ) =
(q ∗ , q ∗ ) and the segregated equilibrium (q1∗ , q2∗ ) = (q ∗ , 0) with q ∗ < 1/2 may be both stable steady states.
The result stems from the fact that urban equilibrium and the cultural composition of the population are co-determined. For some parameters’ values, a given population composition gives rise to
some urban equilibrium which can be either integrated or segregated (see Proposition 3), and in turn
the urban equilibrium provides the cultural environment that enables to preserve this population
composition.7
For instance, if Q0 is low, an integrated equilibrium can emerge (cf. item (ii) of Proposition 3).
This equilibrium arises due to the high probability with which type b agents transmit their trait
in both neighborhoods. Therefore the integrated equilibrium impedes the collapse of population
b. Furthermore, due to cultural substitution, parents of type a exert a socialization effort which
counters the high transmission of trait b and avoids the extinguishment of population a. When these
two forces counterbalance each other, the integrated equilibrium and the associated distribution of
cultural traits in the population can both be sustained over time.
For higher values of Q0 , a segregated equilibrium emerges (again, Proposition 3 states that under
complementarity the urban equilibrium depends on the value of Q). When homogeneity forces,
i.e. conformism and peer effects, are high enough, urban segregation maintains this distribution of
traits over time. In the homogeneous neighborhood, the cultural uniformity remains due to strong
homogeneity forces. In the culturally mixed neighborhood, the incentives of both types parents to
preserve their own trait are such that both cultures persist. In such a case, the segregated equilibrium
and the associated distribution of cultural traits may be sustained in the long run.
Proposition 4 is illustrated in Figure 1. When the intial fraction of majority agents, Q0 , is low
(i.e., Q0 < Q̄) the urban equilbrium is integrated. This equilibrium gives rise to a particular dynamics
of cultural traits which is represented by the bold line. When the dynamics of Qt follows the bold
7
We proceed in the proof by showing that, for some functional forms of transition probabilities, oblique transmission
technology, socialization cost and peer effects, there exists a set of parameters such that both the integrated equilibrium
and segregated equilibrium exist in the long run and are spatially and dynamically stable. Let us mention that the
condition q ∗ < 1/2 is equivalent to item (ii) of Proposition 3 given the particular case we consider in the proof.
18
Qt+1 − Qt
Integrated equilibrium
Segregated equilibrium
Dynamics under integration
Dynamics under segregation
Stable steady state
Qt
Q
Figure 1: Example of the dynamics of cultural traits under cultural proximity
line, there exists a stable stationary equilibirum Q < Q̄ (due to the intuition that given above).
Therefore, for low Q0 , the city can converge to a long run equilibirum which is integrated.
When the initial fraction of majority agent, Q0 , is high (i.e., Q0 > Q̄) the urban equilbrium is
segregated. The dynamics of cultural traits is then represented by the dashed line. In this case,
there exists a stable steady state Q > Q̄. Hence, for higher values of Q0 , the city can converge to a
segregated equilibirum.
Our result is consistent with empirical findings of Card, Mas and Rothstein (2007, 2008) who
show, using Census tract data for the 1970-2000 Censuses, that segregation is not the end of the
city history and integrated neighborhoods are sustainable in the long run.8 However, by contrast to
the approach of Card, Mas and Rothstein (2007, 2008), our model of cultural transmission allows to
shed light on the relationship between the long-run degree of segregation and group share.
Interestingly, Cutler, Glaeser and Vigdor (2008) provide evidence for a substantial variation of
8
In their study of american segregation history, Cutler, Glaeser and Vigdor (1999) document that segregation fell
from 1970 to 1990 throughout the country, providing evidence that there are some forces pushing away from extreme
segregation. From a case studies of segregation in Atlanta, Sacramento and Cleveland, they also stress the diversity of
segregation patterns, with some cities (Atlanta and Cleveland) remaining highly segregated and others (Sacramento)
with sustainable patterns of integration.
19
segregation throughout ethnic groups, space and time. They identify group share as a significant
determinant of the degree of segregation and find that the impact of group share may be positive
or negative depending on the segregation index considered. Groups forming higher shares of the
metropolitan population tend to be both more isolated and less dissimilar than other groups9 . As
pointed out by the authors, this result provides support for larger group spreading out in more
neighborhoods while maintaining a high concentration in other neighborhoods. By contrast, smaller
groups may need to cluster in order to benefit from shared cultural amenities. Our theoretical
framework provides a rationale for this empirical result. Here, individuals from a larger group
have a high chance to transmit their trait whatever the neighborhood they live in. Since they
benefit from living next to mainstream individuals (which is the case because there exist some local
spillovers increasing the prospects of economic success), they have an incentive to spread out in more
neighborhoods, so as to benefit from the local externalities, while still maintaining a high transmission
of their trait. By contrast, individuals from smaller groups do not have this opportunity otherwise
they would reduce drastically the probability to transmit their trait. As a consequence, individuals
from smaller groups prefer to cluster in few areas in order to ensure the persistence of their culture
at the expense of better economic prospects.
An interesting implication of Proposition 4 is that it provides conditions for the efficacy of policies
that would aim to favor integration. Integration policies would more easily help the cultural mixing
to maintain in neighborhoods provided the cultural distance is low and the size of the minority group
is relatively large in the city population.
Finally, results of Section 3 and 4 which both echoe empirical findings by Cutler, Glaeser and
Vigdor (2008) can be summarized in the following graph. Urban equilibrium in the long run, and then
the degree of segregation, depends on both cultural distance and the population composition. The
x-axis captures cultural distance as given by ∆V bb − ∆V ba . When ∆V bb − ∆V ba is negative, there is
cultural polarization, while if positive there is cultural proximity. The y-axis captures the differential
of probabilities P bb − P ba which is determined by the population composition (it is decreasing in
the fraction of type-a agents). When this differential is negative (resp. positive), the distribution of
cultural traits is biased toward the majority (resp. minority) trait.
As illustrated in Figure 2, when there is cultural polarization and/or the population composition
is biased toward the majority cultural trait, the city ends-up segregated. However, when there is
9
Dissimilarity is high when some ethnic group disproportionately resides in some area of a city relative to mainstream group while isolation measures the degree of exposure that individuals of some ethnic group have to other
members of their group (see Cutler, Glaeser, Vigdor, 1999, for more precise definition and specification).
20
Integrated equilibrium
Segregated equilibrium
Pbb − Pba
Segregated equilibrium
Segregated equilibrium
Cultural polarization
Cultural proximity
Cultural distance: ∆Vbb − ∆Vba
Figure 2: Urban equlibria as a function of cultural distance and the population composition
both cultural proximity and the distribution of cultural traits is biased toward the minority, the
long-run city is integrated.
5
Is social segregation optimal?
We now turn to the issue whether it is efficient to let people sort themselves into urban areas, or
should we implement particular urban policies ? In particular, we consider urban policies which affect
the location of agents within the city. Enforcing quotas of inhabitants from a given social category
is one way of promoting social mixing in a given urban area10 . We discuss whether integration or
segregation is more desirable in view of our efficiency criterion, i.e. the long-run rate of education.
Proposition 5 Suppose that Assumption 3 holds and 2q̂ < 1. For any Q0 ∈ [q̂, 2q̂], segregation is
efficient. For any Q0 ∈ [2q̂, 1], integration is efficient.
Proposition 5 stresses that the efficiency of the urban policy depends on the cultural composition
10
One example of a quota policy is the SRU law (loi relative à la Solidarité et au Renouvellement Urbains) in
force in France since 2000. In French municipalities with at least 3,500 inhabitants (1,500 inhabitants in the Paris
administrative region, Ile-de-France), 20% of the available housing stock must be public housing. Municipalities with
figures below this ratio have to pay fines (see Gobillon and Vignolles, 2016, for an evaluation of this policy, and also
Brueckner, 2011, for a presentation and an analysis of various housing policies promoting integration).
21
of the population. The intuition is that, depending on the composition of the city population, either
segregation or integration provides higher incentives to socialize children to cultural trait a which
favors education. For some relatively low initial fractions of the mainstream group, i.e. Q0 ∈ [q̂, 2q̂],
segregation, which concentrates type a individuals in neighborhood 1, allows for sufficient peer effects
in this neighborhood to provide high incentives to transmit cultural trait a compared to trait b. By
way of contrast, were the city to be perfectly integrated, it would reduce the fraction of mainstream
agents, and thus peer effects in neighborhood 1, in such a way that the rate of transmission of trait
b would be higher in both urban areas. This would negatively affects education in the long run.
However, for higher initial fractions of the mainstream group, integration, which increases the
fraction of agents a in neighborhood 2 (compared to segregation), rises the intensity of peer effects
in this neighborhood. When the city fraction of the mainstream group is sufficiently high, i.e.
Q0 ∈ [2q̂, 1], this provides higher incentives to socialize children to trait a. This benefits to education
in the long run. On the contrary, segregation, by reducing the fraction the mainstream inhabitants
in neighborhood 2 would decrease the incentives to transmit trait a in this neighborhood favoring
the spreading of trait b.
Note that due to cultural substitution (cf. Assumption 3), it is never profitable for the long-run
rate of education to have a too large fraction of type a agents, whatever the urban equilibrium.
Incentives to transmit trait a would become low as mainstream parents would rely more intensively
on oblique transmission to transmit their trait.
Whereas Bénabou (1996a) stresses that the degree of complementarity between individuals’ levels of human capital at the community and the society levels is key to assess the efficiency of a
segregated equilibrium, We emphasize the importance of the population cultural composition. Our
result that the efficiency of segregation depends on the distribution of culture in the whole population has important implications. It suggests that poverty deconcentration and integration policies
must circumvent the difficulty to identify the degree of neighborhood social mix most favorable to
education. According to Galster (2002)’s meta-analysis of the empirical evidence on the impact of
poverty concentration on socioeconomic success, if behavioral problems are related to neighborhood
poverty rates within a range of approximately 15-40% of poverty rate,
“This implies that net social benefits will be larger if neighborhoods with greater than
roughly 15% poverty rates are replaced with (an appropriately larger number) of neighborhoods having less than 15% poverty rates. However, net social benefits will be smaller
22
if neighborhoods with greater than about 40% poverty rates are replaced with (an appropriately larger number) of neighborhoods having between about 15-40% poverty rates.
Put more bluntly in policy terms, unless very low-poverty neighborhoods can be opened
up for occupation by the poor, deconcentration efforts should halt, because merely transferring the poor from high- to moderate-poverty neighborhoods is likely to be socially
inefficient.” (p. 322, Galster, 2002)
What are the consequences of urban policies on economic inequalities? To answer that question,
let us compare the educational gaps under integration and segregation. Suppose that the city is
integrated at date t, i.e. qt1 = qt2 = Qt /2, given (12) the educational gap, is given by
P
ē|a
Qt
2
−P
ē|b
Qt
2
.
When the city is segregated, the gap equals
P e|a − P e|b =





ē|a
P1 (Qt ) −
ē|a
P1 (1)
Qt
+
Qt −1 ē|a
P2
Qt
ē|b
ē|b
(1−Qt )P1 (Qt )+P2 (0)
(2−Qt )
if Qt ≤ 1
ē|b
(Qt − 1) − P2 (Qt − 1),
otherwise.
It is easy to check that under Assumption 1, educational gap is lower under integration than under
segregation. Segregation increases the cultural disparity between neighborhoods which magnifies
peer effects differences between both traits. It turns out that, for some cultural composition of the
population, there is a trade-off between efficiency and equity: while segregation promotes long-run
education, it widens the educational gap between the two cultural groups.
6
Conclusion
How does segregation impact the transmission of traits which are critical for economic success?
When does cultural heterogeneity leads to residential segregation? This paper provides some answer
by developing a model where neighborhood formation and cultural transmission interact.
A key feature of our framework is that the parental choice of the place of residence relies on the
need to socially integrate and the concern to preserve own culture. This tension between culture and
economic integration impacts cultural diversity and residential segregation in the long run.
When the minority culture shares the mainstream view that education is valuable, multiple types
of long-run urban configurations arise depending on initial cultural composition of the society. In
23
particular, an integrated city can emerge in the long run provided that cultural mixing in neighborhoods allows for the preservation of the minority culture. We also show that segregation strengthens
the parental influence on the child’s destiny. Finally, we highlight the crucial role of the population
composition to assess the efficiency of urban policies.
The model could be extended along several lines. First, there is a consensus that housing market
dynamics impact segregation (see the review of Rosenthal and Ross, 2014). The model is flexible
enough to introduce some housing market features such as tenure choice, housing depreciation and
maintenance, development and redevelopment of housing stock. These features would influence locations choices and allow us to explore their implications on the pattern of segregation and cultural
dynamics. Second, our model considers that cultural distance is binary: there is either cultural
proximity or cultural polarization. Integrating identity behaviors that can lead to cultural clash or
cultural assimilation would allow us to endogenize cultural distance and study how it responds to
assimilation policies. Third, relaxing the assumption that children attend the school of their urban
area would allow to differentiate the social arenas where peer effects and oblique transmission are
determined. For instance, considering that peer effects are circumscribed within schools whereas
oblique transmission is produced in the urban area would affect the trade-off faced by parents when
deciding the place of residence. This extension could shed new light on the consequences of school
choice systems on segregation and inequality dynamics. Finally, the literature on cultural transmission has addressed the issue of the design and dynamics of institutions (see Bisin and Verdier,
2015). However, this literature disregards endogenous stratification. Fruitful research would be to
investigate how culture interacts with both residential segregation and institutions providing local
public goods and education services.
References
[1] Akerlof, G., 1980, “A Theory of Social Custom of Which Unemployment May Be One Consequence”, Quarterly Journal of Economics, vol. 94, 749-775.
[2] Akerlof, G. and R. Kranton, 2002, “Identity and Schooling: Some Lessons for the Economics of
Education”, Journal of Economic Literature, vol. 15, 1167-1201.
[3] Anderson, E., 1999, The Code of The Street: Decency, Violence, and the Moral Life of the
Inner-City, New York: W.W. Norton & Company.
24
[4] Battu, H. and Y. Zenou, 2010, “Oppositional Identities and Employment for Ethnic Minorities.
Evidence from England,” Economic Journal, vol.120, 52-71.
[5] Battu, H., McDonald, M. and Y. Zenou, 2007, “Oppositional Identities and the Labor Market,”
Journal of Population Economics, vol. 20, 643-667.
[6] Bénabou, R., 1993, “Workings of a City: Location, Education, and Production”, Quarterly
Journal of Economics, vol. 108, 619-652.
[7] Bénabou, R., 1996a, “Heterogeneity, Stratification and Growth: Macroeconomic Implications of
Community Structure and School Finance”, American Economic Review, vol. 86, 584-609.
[8] Bénabou, R., 1996b, “Equity and Efficiency in Human Capital Investment: the Local Connection”, Review of Economic Studies, vol. 63, 237-264.
[9] Bisin, A. and Th. Verdier, 2001, “The Economics of Cultural Transmission and the Evolution
of Preferences”, Journal of Economic Theory, vol. 97, 298-319.
[10] Bisin, A. and Th. Verdier, 2015, “On the Dynamics of Culture and Institutions”, mimeo.
[11] Bisin, A., Topa, G. and Th. Verdier, 2004, “Religious Intermarriage and Socialization in the
United States”, Journal of Political Economy, vol. 112, 615-664.
[12] Bisin, A., Patacchini, E., Verdier, Th. and Y. Zenou, 2011a, “Ethnic Identity and Labor-Market
Outcomes of Immigrants in Europe”, Economic Policy, vol. 65, 57-92.
[13] Bisin, A., Patacchini, E., Verdier, Th. and Y. Zenou, 2011b, “Formation and Persistence of
Oppositional Identities”, European Economic Review, vol. 55, 1046-1071.
[14] Borjas, G., 1995, “Ethnicity, Neighborhoods, and Human-Capital Externalities”, American Economic Review, vol. 85, 365-390.
[15] Borjas, G., 1998, “To Ghetto or Not to Ghetto: Ethnicity and Residential Segregation”, Journal
of Urban Economics, vol. 44, 228-253.
[16] Botticini, M. and Z. Eckstein, 2005, “Jewish Occupational Selection: Education, Restrictions,
or Minorities?”, Journal of Economic History, vol. 65, 922-948.
25
[17] Botticini, M. and Z. Eckstein, 2007, “From Farmers to Merchants, Conversions and Diaspora:
Human Capital and Jewish History”, Journal of European Economic Association, vol. 5, 885926.
[18] Brueckner, J., 2011, Lectures on Urban Economics, The MIT Press, Cambridge, Massachusetts.
[19] Card, D., Mas, A. and J. Rothstein, 2007, “Tipping and the Dynamics of Segregation”, Quarterly
Journal of Economics, vol. 123, 177-218.
[20] Card, D., Mas, A. and J. Rothstein, 2008, “Are Mixed Neighborhoods Always Unstable? TwoSided and One-Sided Tipping”, NBER Working Paper 14470.
[21] Carvalho, J-P., 2013, “Veiling”, Quarterly Journal of Economics, vol. 128, 337-370.
[22] Chetty, R., Hendren, N. and L. Katz, 2015, “The Effects of Exposure to Better Neighborhoods
on Children: New Evidence from the Moving to Opportunity Experiment”, mimeo.
[23] Chetty, R., Hendren, N., Kline, P. and E. Saez, 2014, “Where is the Land of Opportunity? The
Geography of Intergenerational Mobility in the United States”, Quarterly Journal of Economics,
vol. 129, 1553-1623.
[24] Constant, A., S., Schüller, and K. Zimmermann, 2013, “Ethnic Spatial Dispersion and Immigrant Identity”, IZA DP n◦ 7868.
[25] Cutler, D., Glaeser, E. and J. Vigdor, 1999, “The Rise and Decline of the American Ghetto”,
to. Journal of Political Economy, vol. 107, 455-506.
[26] Cutler, D., Glaeser, E. and J. Vigdor, 2008, “Is the Melting Pot Still Hot? Explaining the
Resurgence of Immigrant Segregation”, Review of Economics and Statistics, vol. 90, 478-497.
[27] de Bartolome, Ch., 1990, “Equilibrium and Inefficiency in a Community Model with Peer Effects”, Journal of Political Economy, vol. 98, 110-133.
[28] Doepke, M. and F. Zilibotti, 2008, “Occupational Choice and the Spirit of Capitalism”, Quarterly
Journal of Economics, vol. 123, 747-793.
[29] Domingues Dos Santos, M. and F.C., Wolff, 2011, “Human Capital Background and the Educational Attainment of Second-Generation Immigrants in France”, Economics of Education
Review, vol. 30, 1085-1096.
26
[30] Duranton, G. and D., Puga, 2015, “Urban land use”, in G. Duranton, V. Henderson and W.
Strange (eds), Handbook of Regional and Urban Economics, Amsterdam: Elsevier Publisher.
[31] Durlauf, S., 1996, “A Theory of Persistent Income Inequality”, Journal of Economic Growth,
vol. 1, 75-93.
[32] Durlauf, S., 2004, “Neighborhood Effects”, in J. V. Henderson and J. F. Thisse (eds), Handbook
of Regional and Urban Economics, Amsterdam: Elsevier Publisher.
[33] Dustmann, Ch., Frattini, T. and G. Lanzara, 2012, “Educational Attainment of SecondGeneration Immigrants: An International Comparison”, Economic Policy, vol. 27, 143-185.
[34] Fouka, V., 2016, “Backlash: The Unintended Effects of Language Prohibition in US Schools
after World War I”, mimeo.
[35] Fryer, R. and P. Torelli, 2010, “An Empirical Analysis of Acting White”, Journal of Public
Economics, vol. 94, 380-396.
[36] Galster, G., 2002, “An Economic Efficiency Analysis of Deconcentrating Poverty Populations”,
Journal of Housing Economics, vol. 11, 303-329.
[37] Gang, I. N. and K. F. Zimmermann, 2000, “Is Child Like Parent? Educational Attainment and
Ethnic Origin”, Journal of Human Resources, vol. 35, 550-569.
[38] Gobillon, L. and B. Vignolles, 2016, “Evaluation de l’effet d’une politique spatialisée d’accès au
logement : la loi SRU”, La Revue Economique, vol. 67, 615-638.
[39] Guiso, L., Sapienza, P. and L. Zingales, 2008, “Social Capital as Good Culture”, Journal of the
European Economic Association, vol. 6, 295-320.
[40] Ioannides, Y. and G. Zanella, 2008, “Searching for the Best Neighborhood: Mobility and Social
Interactions”, mimeo.
[41] Kling, J. R., Liebman, J. B. and L. F. Katz, 2007, “Experimental Analysis of Neighborhood
Effets”, Econometrica, vol. 75, 83-119.
[42] Lamont, M. and M. L. Small, 2008, “How Culture Matters for the Understanding of Poverty:
Enriching our Understanding”, in L. A. Chih and D. R. Harris (eds), The Colours of Poverty:
Why Racial and Ethnic Disparities Exist, New York: Russell Sage Foundation.
27
[43] Lindbeck, A. and S. Nyberg, 2006, “Raising Children to Work Hard: Altruism, Work Norms,
and Social Insurance”, Quarterly Journal of Economics, vol. 121, 1473-1503.
[44] Loury, G., 1977, “A Dynamic Theory of Racial Income Differences”, in A. Le Mond (eds),
Women, Minorities and Employment Discrimination, Lexington, MA: Lexington Books.
[45] Moizeau, F., 2015, “Dynamics of Social Norms in the City”, Regional Science and Urban Economics, vol. 51, 70-87.
[46] Oreopoulos, P., 2003, “The Long-Run Consequences of Living in a Poor Neighborhood”, Quarterly Journal of Economics, vol. 118, 1533-1575.
[47] Rosenthal, S. and and S. Ross, 2014, “Change and Persistence in the Economic Status of Neighborhoods and Cities”, in G. Duranton, V. Henderson and W. Strange (eds), Handbook of Regional
and Urban Economics, Amsterdam: Elsevier Publisher.
[48] Saez-Marti, M., and A. Sjögren, 2008, “Peers and Culture”, Journal of Economic Theory, vol.
143, 153-176.
[49] Saez-Marti, M. and Y. Zenou, 2012, “Cultural Transmission and Discrimination ”, Journal of
Urban Economics, vol. 72, 137-146.
[50] Sharkey, P., 2008, “The Intergeneration Transmission of Context”, American Journal of Sociology, vol. 113, 931-969.
[51] Sharkey, P. and F. Elwert, 2011, “The Legacy of Disadvantage: Multigenerational Neighborhood
Effects on Cognitive Ability”, American Journal of Sociology, vol. 116, 1934-1981.
[52] Sikkink, D. and M.O. Emerson, 2008, “School Choice and Racial Segregation in US Schools:
The Role of Parents’ Education”, Ethnic and Racial Studies, vol. 31, 267-293.
[53] Tinker, C. and A. Smart, 2012, “Constructions of Collective Muslim Identity by Advocates of
Muslim Schools in Britain”, Ethnic and Racial Studies, vol. 35, 643-663.
[54] Topa, G. and Y. Zenou, 2014, “Neighborhood versus Network Effects”, in G. Duranton, V.
Henderson and W. Strange (eds), Handbook of Regional and Urban Economics, Amsterdam:
Elsevier Publisher.
28
[55] Verdier, Th. and Y. Zenou, 2017, “The Role of Social Networks in Cultural Assimilation”,
Journal of Urban Economics, vol. 97, 15-39.
[56] Weber, Max, 1958, The Protestant Ethic and the Spirit of Capitalism, New York: Charles
Scribner’s Sons.
[57] Wilson, W.J., 1987, The Truly Disadvantaged: The Inner City, the Urban Underclass and Public
Policy, Chicago: University of Chicago Press.
[58] Wodtke, G.T., Harding, D.J. and F. Elwert, 2011, “Neighborhood Effects in Temporal Perspective: The Impact of Long-Term Exposure to Concentrated Disadvantage on High School
Graduation”, American Sociological Review, vol. 76, 713-736.
29
7
Appendix
7.1
Socialization Choice of Trait b Parents
At date t, a trait-b parent chooses her optimal effort τ ∗b that solves
max U b (ρ, τ b ) subject to (2).
τb
This leads to the following first-order condition:
f (q)(P ē|b ∆V bb − P ē|a ∆V ba + V e|bb − V e|ba ) = Θ0 (τ ∗b )
(15)
We assume that for any q
P ē|b ∆V bb − P ē|a ∆V ba + V e|bb − V e|ba ≥ 0.
(16)
which amounts to say that trait-b parents have an incentive to transmit their own trait, i.e. τ ∗b (q) ≥ 0.
By the implicit function theorem, we get
ē|a
f 0 (q) P ē|b ∆V bb − P ē|a ∆V ba + V e|bb − V e|ba + f (q) dPdq ∆V bb − ∆V ba
dτ ∗b
=
.
dq
Θ00 (τ ∗b )
7.2
(17)
Bid-Rent Slopes
Using the envelope theorem, the slope of the trait-a bid curve is given as
dρa
ē|a
ē|b
= (1 − τ ∗a (q1 ))f 0 (q1 )(P1 ∆V aa − P1 ∆V ab + V e|aa − V e|ab )
dq1
dq2 ē|a
ē|b
−(1 − τ ∗a (q2 ))f 0 (q2 )
P2 ∆V aa − P2 ∆V ab + V e|aa − V e|ab
dq1
ē|a
aa
aa
aa
ab dP1
+ P1 ∆V + (1 − P1 )∆V
e
dq1
dP2ē|a dq2
− P2aa ∆V aa + (1 − P2aa )∆V ab
.
dq2 dq1
30
(18)
For type b individuals, we have
dρb
ē|b
ē|a
= −(1 − τ ∗b (q1 ))f 0 (q1 )(P1 ∆V bb − P1 ∆V ba + V e|bb − V e|ba )
dq1
dq2 ē|b
ē|a
(P2 ∆V bb − P2 ∆V ba + V e|bb − V e|ba )
+(1 − τ ∗b (q2 ))f 0 (q2 )
dq1
dP1ē|b
+ P1bb ∆V bb + 1 − P1bb ∆V ba
dq1
dP2ē|b dq2
.
− P2bb ∆V bb + (1 − P2bb )∆V ba
dq2 dq1
7.3
(19)
Proof of Proposition 1
Given (18) and (19), and using dq2 /dq1 = −1, the bid-rent slope differential for any given q1 is
dρa dρb
ē|a
ē|b
−
= (1 − τ ∗a (q1 ))f 0 (q1 )(P1 ∆V aa − P1 ∆V ab + V e|aa − V e|ab )
dq1
dq1
ē|b
ē|a
ē|b
ē|a
+(1 − τ ∗b (q1 ))f 0 (q1 )(P1 ∆V bb − P1 ∆V ba + V e|bb − V e|ba )
ē|a
ē|b
e|ab
∗a
0
aa
ab
e|aa
−V
+(1 − τ (q2 ))f (q2 ) P2 ∆V − P2 ∆V + V
+(1 − τ ∗b (q2 ))f 0 (q2 )(P2 ∆V bb − P2 ∆V ba + V e|bb − V e|ba )
ē|a
ē|b
dP1
dP
∆V aa + (1 − P1aa ) 1 ∆V ab
dq1
dq1
ē|b
ē|a
bb dP1
bb
bb dP1
−P1
∆V − 1 − P1
∆V ba
dq1
dq1
+P1aa
ē|a
ē|b
aa dP2
aa
aa dP2
+P2
∆V + (1 − P2 )
∆V ab
dq2
dq2
ē|b
ē|a
dP
dP
−P2bb 2 ∆V bb − (1 − P2bb ) 2 ∆V ba .
dq2
dq2
(20)
First, let us consider that ∆V ab = α∆V aa with α ∈ [0, 1] . One can see that the bid-rent slope
differential is, other things being equal, a linear function of ∆V aa with slope
ē|a
ē|b
ē|a
ē|b
(1 − τ ∗a (q1 ))f 0 (q1 ) P1 − αP1 + (1 − τ ∗a (q2 ))f 0 (q2 ) P2 − αP2
!
!
ē|a
ē|a
ē|b
ē|b
dP
dP
dP
dP
+ P1aa 1 + (1 − P1aa ) 1 α + P2aa 2 + (1 − P2aa ) 2 α
dq1
dq1
dq2
dq2
which is positive. Hence, one could choose ∆V aa high enough so that dρa /dq1 >dρb /dq1 . This
amounts to say that if mainstream parents attach much more importance to education than the
minority ones whatever the type of the child (∆V aa > ∆V ab > ∆V ba > ∆V bb ), then segregation
arises. As we aim to capture the key role of cultural distance, we do not put emphasis on this case
31
in the main text eventhough it could arise in practice.
Let us consider now that ∆V aa > ∆V ab , ∆V ba > ∆V bb , ∆V aa = ∆V ba and ∆V ab = ∆V bb . Due to
the oblique transmission effect, the first four terms of (20) are positive. Hence, a sufficient condition
for the bid-rent slope differential to be positive is that the following inequality is satisfied
ē|b
ē|a
P1aa + P1bb − 1
dP
dP1
∆V ba − 1 ∆V bb
dq1
dq1
ē|a
!
ē|b
dP2
dP
∆V ba − 2 ∆V bb
dq2
dq2
+ P2aa + P2bb − 1
!
> 0.
Given (1) and (2), we have
ē|a
ē|b
dP1
dP
∆V ba − 1 ∆V bb
dq1
dq1
b
τ1a (1 − q1 ) + q1 τ1
ē|a
+
τ2a (1
− q2 ) +
q2 τ1b
ē|b
!
dP
dP2
∆V ba − 2 ∆V bb
dq2
dq2
!
> 0.
Given Assumption 1 and if ∆V ba > ∆V bb , we have any j = 1, 2
ē|a
dPj
dqj
∆V ba − ∆V bb > 0 for j.
Hence, the segregated equilibrium exists.
Spatial stability requires that, after a move of a small number of trait-a individuals from neighborhood 1 to neighborhood 2 and a migration of the same number of trait-b individuals in the reverse
direction, the highest bidders for neighborhood 1 are trait-a individuals. At the segregated equilibrium when Q < 1, q1∗ = Q and q2∗ = 0, the bid-rent equilibrium is the willingness of trait-b individuals.
At this equilibrium price level, trait-a individuals strictly prefer to live in urban area 1. When Q ≥ 1,
q1∗ = 1 and q2∗ = Q − 1, the bid-rent equilibrium is the willingness of trait-a individuals. At this
equilibrium price level, trait-b individuals strictly prefer to live in urban area 2. Whatever one of
these equilibrium configurations, trait-a individuals remain the highest bidders for neighborhood 1
after the small perturbation of the equilibrium.
32
7.4
Proof of Corollary 1
We consider the following functional forms: P ē|i = αi q, with αa > αb , f (q) = q and Θ(τ ) = (1/2θ) τ 2 .
From (5) and (15), we get
τ ∗a = −θq 2 (αa ∆V aa − αb ∆V ab ) + θq((αa ∆V aa − αb ∆V ab ) − (V e|aa − V e|ab )) + θ(V e|aa − V e|ab ),
and
τ ∗b = θq 2 (αb ∆V bb − αa ∆V ba ) + θq(V e|bb − V e|ba ).
Let us consider the probability differential
q1 ∗a
Q − q1 ∗a
(τ (q1 ) + (1 − τ ∗a (q1 ))f (q1 )) +
(τ (Q − q1 ) + (1 − τ ∗a (Q − q1 ))f (Q − q1 ))
Q
Q
1 + q1 − Q
1 − q1
−
(1 − τ ∗b (q1 ))f (q1 ) −
(1 − τ ∗b (Q − q1 ))f (Q − q1 ) .
2−Q
2−Q
P aa − P ba =
Let us compare this differential under segregation and integration. Suppose that Q < 1 (similar
arguments hold for the case Q > 1 so that we skip the proof in order to lighten the exposition). The
differential of probabilities at the segregated equilibrium writes as
P aa − P ba = (τ ∗a (Q) + (1 − τ ∗a (Q))Q) −
1−Q
(1 − τ ∗b (Q))Q.
2−Q
(21)
Under integration, we have
P
aa
−P
ba
Q
Q
Q
Q
Q
∗a
∗a
∗b
= τ
+ 1−τ
− 1−τ
.
2
2
2
2
2
(22)
We have a higher differential under segregation if (21) is greater than (22) that is
Q
Q
Q
∗a
∗a
+ 1−τ
(τ (Q) + (1 − τ (Q))Q) − τ
2
2
2
1−Q
Q
Q
>
(1 − τ ∗b (Q))Q − 1 − τ ∗b
,
2−Q
2
2
∗a
∗a
Since the function τ ∗b is increasing in q, we have (1 − τ ∗b (Q)) < 1 − τ ∗b
if
1−Q
Q
Q − ≤ 0,
2−Q
2
33
Q
2
. The RHS is negative
which is equivalent to
2Q ≥ Q,
which is true. Therefore, if the LHS is positive then (21) is greater than (22). The LHS is equal to
zero at Q = 0. If P aa = τ ∗a (Q) + (1 − τ ∗a (Q))Q is an increasing function of Q, then the LHS is
strictly positive for any Q > 0. Let us perform the derivative, one gets
dP aa
= −2θ(αa ∆V aa − αb ∆V ab )(1 − q) + θ(αa ∆V aa − αb ∆V ab − (V e|aa − V e|ab ))(1 − 2q)
dq
+ θq 2 (αa ∆V aa − αb ∆V ab ) − θ(V e|aa − V e|ab ) + 1.
This function is a polynomial of order two which is convex. It is equal to 1 at q = 1. It is positive
at q = 0 if and only if
−θ(αa ∆V aa − αb ∆V ab ) − 2θ(V e|aa − V e|ab ) + 1 ≥ 0
leading to
θ≤
2(V
e|aa
−V
e|ab )
1
≡ θe1 .
+ (αa ∆V aa − αb ∆V ab )
The derivative of the polynomial dP aa /dq is positive at q = 0 so that we conclude that when the
above inequality holds, dP aa /dq is positive on the whole interval [0, 1]. Hence, the result.
7.5
Proof of Proposition 2
1. From (8), let us study the dynamics described by the map Ψ : [0, 1] → [0, 1] defined such that
Ψ(qt ) = (f (qt ) − qt )(1 − τ b (qt )) + (τ a (qt ) − τ b (qt ))qt (1 − f (qt )) + qt .
Steady states are such that Ψ(qt ) − qt = 0. First, we have Ψ(0) − 0 = 0, Ψ(1) − 1 = 0 so that
q = 0 and q = 1 are steady states of the map Ψ.
0
Let us perform the derivative Ψ , we obtain
dτ b
+
Ψ (qt ) =(f (qt ) − 1)(1 − τ (qt )) − (f (qt ) − qt )
dqt
0
0
b
dτ a dτ b
−
dqt
dqt
0
qt (1 − f (qt ))
+ (τ a (qt ) − τ b (qt ))(1 − f (qt )) − (τ a (qt ) − τ b (qt ))qt f (qt ) + 1.
34
We deduce,
0
0
0
0
Ψ (0) = τ a (0) + f (0) > 0,
and
Ψ (0) = τ b (1) + f (1) > 0.
0
0
With item (i) and (ii) of Assumption 3 we have Ψ (0) < 1, Ψ (1) > 1. Also, with item (iii), we known
that there exists some q ∈]0, 1[ such that Ψ(q) > q. As we have Ψ ∈ C 2 , we deduce that there exist
0
0
q̂, q ∗ , q̂ < q ∗ , such that (i) Ψ(q̂) = q̂, Ψ(q ∗ ) = q ∗ , Ψ (q̂) > 1, Ψ (q ∗ ) < 1.
2. Cultural dynamics in the segregated city.
The dynamics of cultural traits in the city is described by the map Q : [0, 2] → [0, 2] which is
such that
1
2
Q(Qt+1 ) = Q(qt+1
+ qt+1
) = Ψ(qt1 ) + Ψ(qt2 ).
Fixed points of the map Q are such that
Ψ(qt1 ) + Ψ(qt2 ) = qt1 + qt2 .
When the city is segregated, steady states are Q = 0, Q = q̂, Q = q ∗ , Q = 1, Q = 1 + q̂,
Q = 1 + q ∗ , Q = 2 with urban urban equilibria respectively given by (q 1 , q 2 ) = (0, 0), (q 1 , q 2 ) = (q̂, 0),
(q 1 , q 2 ) = (q ∗ , 0), (q 1 , q 2 ) = (1, 0), (q 1 , q 2 ) = (1, q̂), (q 1 , q 2 ) = (1, q ∗ ), (q 1 , q 2 ) = (1, 1).
A necessary and sufficient condition for stability of any steady state Q when the segregated urban equilibrium is (q 1 , q 2 ) is11 .
d(Qt+1 − Qt )
|Q < 0,
dQt
0
0
⇔Ψ (q 1 ) + Ψ (q 2 ) < 2.
0
Since item (i) of Assumption 3 implies Ψ (0) < 1 we deduce that the equilibrium (0, 0) is stable.
0
0
0
Furthermore, from the previous part of this proof, we have Ψ (q ∗ ) < 1 so that Ψ (0) + Ψ (q ∗ ) < 2 and
we deduce that the equilibrium (q ∗ , 0) is stable. Now, from item (iii) of Assumption 3, we also have
11
This is also a sufficient condition for stability because one can easily check that
35
dQt+1
dQt
> 0.
0
0
0
0
0
0
Ψ (1) > 1, we deduce that Ψ (1) + Ψ (1) > 2 and Ψ (1) + Ψ (q̂) > 2 (since Ψ (q̂) > 1). Equilibria
(1, 1) and (1, q̂) are unstable. For equilibria (q̂, 0), (1, 0), and (1, q ∗ ), our assumptions do not allow to
conclude. We deduce that the population dynamics in the segregated city admits at least to stable
long-run equilibria Q = 0 and Q = q ∗ .
7.6
Proof of Proposition 3
Existence. When q1 = q2 = Q/2, both willignesses to pay are equal to 0 implying that nobody
has an incentive to move. The integrated city, where both neighborhood cultural compositions are
identical, is a urban equilibrium.
Spatial stability. The stability condition requires that after a move of a small number of trait-b
individuals from neighborhood 1 to neighborhood 2 and a migration of the same number of trait-a
individuals in the reverse direction, the highest bidders for neighborhood 1 are trait-b individuals.
This amounts to check whether the bid rent is steeper for the trait b-individuals. Formally, stability
requires that
dρb dρa −
< 0.
dq1 q∗ =Q/2 dq1 q∗ =Q/2
1
1
ē|j
At the integrated equilibrium, as q1∗ = q2∗ = Q/2, we have P1aa = P2aa , P1bb = P2bb , dP1 /dq1 =
ē|j
− dP2 /dq1 · (dq1 /dq2 ) for j = a, b and −f 0 (q1 ) = f 0 (q2 )(dq2 /dq1 ). Hence, the bid-slope rent
differential becomes
dρa dρb Q
Q
ē|a
ē|b
∗a
−
= 2 1−τ
f0
(P1 ∆V aa − P1 ∆V ab + V e|aa − V e|ab )
dq1 q∗ =Q/2 dq1 q1 =Q/2
2
2
1
Q
Q
ē|b
ē|a
+2 1 − τ ∗b
f0
(P1 ∆V bb − P1 ∆V ba + V e|bb − V e|ba )
2
2
ē|a
ē|b
aa dP1
aa
aa dP1
+2P1
∆V + 2 (1 − P1 )
∆V ab
dq1
dq1
ē|b
dP1ē|a
dP
−2P1bb 1 ∆V bb − 2 1 − P1bb
∆V ba .
dq1
dq1
Given the symmetry assumption, ∆V aa = ∆V bb and ∆V ab = ∆V ba and Assumption 1 the bid-rent
36
slope differential can be written as follows:
Q
dρa Q
dρb ē|a
ē|b
0
∗a
f
; r, γ (P1 ∆V aa − P1 ∆V ab + V e|aa − V e|ab )
−
= 2 1−τ
dq1 q∗ =Q/2 dq1 q1 =Q/2
2
2
1
Q
Q
ē|b
ē|a
0
∗b
f
; r, γ (P1 ∆V bb − P1 ∆V ba + V e|bb − V e|ba )
+2 1 − τ
2
2
ē|a
dP1
∆V bb − ∆V ba .
+2 P1aa − P1bb
dq1
Considering that the first two terms being positive due to cultural transmission motives, a necessary
condition for the bid-rent slope differential to be negative is
P1aa − P1bb
∆V bb − ∆V ba < 0.
If P1aa − P1bb > 0, it is equivalent to
∆V bb − ∆V ba < 0
which is impossible as ∆V bb − ∆V ba < 0 is a sufficient condition for segregation to emerge. We then
must have P1aa − P1bb < 0 and ∆V bb − ∆V ba > 0.
7.7
Proof of Proposition 4
We provide an analytical example for which both types of urban equilibria, integrated equilibrium
and segregated equilibrium, are spatially and dynamically stable in the long run. Let us consider
the following specifications P ē|a = βq + a, P ē|b = βq + ba with a > b, β, a, b > 0 and β + a < 1. Let
Θ(τ ) = θτ 2 /2, and f (qt ) = γrqtα /(rqtα + (1 − qt )α ) with α, r, γ > 0. Let us stress that, in order to be
able to solve the problem analytically, one needs to restrict the model parameters so as to a obtain
closed form solution. The parameter γ is then necessary to keep some degree of liberty. It is not necessary, however, for the result exposed in Proposition 4. A numerical example is available upon request.
1. Stationary distribution of cultural traits and dynamic stability. Given the above functional forms, (5) and (15) can be written as follows
1
τ ∗a (qt ) = (1 − f (qt )) βqt (∆V aa − ∆V e|ab ) + a∆V aa − b∆V ab + V e|aa − V e|ab ,
θ
1
τ ∗b (qt ) = f (qt ) βqt (∆V aa − ∆V ab ) + b∆V aa − a∆V ab + V e|bb − V e|ba .
θ
37
Steady states are such that Qt+1 = Qt which is true if q1,t+1 − q1,t = 0 and q2,t+1 − q2,t = 0. Given
(8) and the above functional forms, for j ∈ {1, 2}, qj,t+1 − qj,t = 0 admits at least three solutions: 0,
1 and q ∈]0, 1[ such that
h(qt ) = 0,
where h(qt ) = f (qt ) − qt + qt τ ∗a (qt ) − f (qt )τ ∗b (qt ) + qt f (qt )(τ ∗b (qt ) − τ ∗a (qt )).
To obtain a closed form solution, let us consider that parameters are such that there exists q ∗ solving
τ ∗a (q ∗ ) − τ ∗b (q ∗ ) = 0,
and
f (q ∗ ) − q ∗ = 0,
which implies h(q ∗ ) = 0. This is equivalent to say that parameters are such that
C1: (1 − q ∗ ) βq ∗ (∆V aa − ∆V ab ) + a∆V aa − b∆V ab + V e|aa − V e|ab −
q ∗ βq ∗ (∆V aa − ∆V ab ) + b∆V aa − a∆V ab + V e|bb − V e|ba = 0,
C2: γ = q ∗ +
q ∗ (1 − q ∗ )α
.
rq ∗ α
Without loss of generality, let us normalize V e|aa − V e|ab = 0. The expression for q ∗ is then
β(∆V aa − ∆V e|ab ) − (∆V aa − ∆V ab )(a + b) − (V e|bb − V e|ba ) +
q∗ =
4β(∆V aa − ∆V ab )
2
with D = (∆V aa − ∆V ab )(b + a − β) + (V e|bb − V e|ba )
√
D
,
+ 8β(∆V aa − ∆V ab )(a∆V aa − b∆V ab ).
Hence, both urban configurations (q1 , q2 ) = (q ∗ , q ∗ ) and (q1 , q2 ) = (0, q ∗ ) are steady states. Let
us now check dynamic stability. Let us first consider the integrated equilibrium (q1 , q2 ) = (q ∗ , q ∗ ). It
is a stable steady state if and only if
dh(q) <0
dq q=q∗
which is equivalent to
0
(1 − τ ∗a (q ∗ ))(f (q ∗ ) − 1) + q ∗ (1 − q ∗ )
38
!
dτ ∗a dτ ∗b −
< 0.
dq q=q∗
dq q=q∗
0
Let us perform f (qt ). We obtain
(α−1)
rq
(1 − qt )(α−1)
f (qt ) = αγ t α
.
(rqt + (1 − qt )α )2
0
0
Assume that α > 1 meaning that there is conformism. In such a case, note that we have f (0) = 0.
We have
0
f (q ∗ ) =
α γ − q∗
γ 1 − q∗
using f (q ∗ ) = q ∗ .
0
0
When γ = 1, f (q ∗ ) = α > 1. When γ equals q ∗ (implying that r is large), f (q ∗ ) = 0. We can
0
deduce that there exists γ̃ such that ∀γ < γ̃, we have f (q ∗ ) < 1.
0
0
Suppose that γ < γ̃. Going back to the sign of h (q ∗ ), given that we have f (q ∗ ) < 1, we deduce
the following.
∗b ∗a 0
− dτdq < 0, then h (q ∗ ) < 0.
(i) If dτdq ∗
∗
q=q
q=q
∗a dτ ∗a dτ ∗b (ii) If dq − dq > 0, noting that dτdq ∗
∗
q=q
q=q
q=q ∗
−
dτ ∗b dq q=q ∗
and τ ∗a (q ∗ ) are proportional to
1/θ, then, for θ sufficiently high both terms become sufficiently low so that the inequality holds. In
e h0 (q ∗ ) < 0.
other words, there exists θe such that ∀θ > θ,
0
Second, let us consider the segregated equilibrium (q1 , q2 ) = (q ∗ , 0). Given that h (q ∗ ) < 0, the
urban configuration (q1 , q2 ) = (q ∗ , 0) is a stable steady state if and only if
0
h (0) < 0 ⇔ −1 + τ a (0) < 0.
which is true.
2. Spatial stability of the integrated equilibrium (q1 , q2 ) = (q ∗ , q ∗ ).
Following proof of Proposition 3, a necessary condition for spatial stability of the integrated
equilibrium is
ē|a
ē|b
P1aa dP1 /dq1 − P1bb dP1 /dq1 < 0
39
which is equivalent to
ē|a
ē|b
dP
dP
(τ + (1 − τ )q ) 1 − (τ ∗ + (1 − τ ∗ )(1 − q ∗ )) 1 < 0.
dq1
dq1
∗
ē|a
∗
∗
ē|b
Since dP1 /dq1 >dP1 /dq1 , this inequality holds only if
(τ ∗ + (1 − τ ∗ )q ∗ ) − (τ ∗ + (1 − τ ∗ )(1 − q ∗ )) < 0
leading to
1
q∗ < .
2
Given our functional forms, and τ a (q ∗ ) = τ b (q ∗ ) = τ ∗ , the condition for the integrated equilibrium
(q1 , q2 ) = (q ∗ , q ∗ ) to be spatially stable, can be written as follows
0
f (q ∗ ) 2β(∆V aa − ∆V ab )q ∗ + (a + b)(∆V aa − ∆V ab ) + V e|bb − V e|ba
+ (∆V aa − ∆V ab )β(2q ∗ − 1) < 0.
0
0
Since we have f (q ∗ ) = 0 when γ = q ∗ , as long as q ∗ < 12 , we deduce that there exists γ such that
0
∀γ < γ , this inequality holds.
3. Existence of the segregated equilibrium (q1 , q2 ) = (q ∗ , 0).
The condition for the segregated equilibrium (q1 , q2 ) = (q ∗ , 0) to exist is ρa (q ∗ , 0) > ρb (q ∗ , 0).
Given the above functional forms, we get
(a∆V aa − b∆V ab )(τ ∗ + (1 − τ ∗ )q ∗ − τ a (0))
+ (V e|bb − V e|ba + b∆V aa − a∆V ab )(1 − τ ∗ )q ∗
+(1 − τ ∗ )βq ∗ (2q ∗ − 1) ∆V aa − ∆V ab > 0.
40
Let us define
Λ(τ ∗ ) ≡ (a∆V aa − b∆V ab )(τ ∗ + (1 − τ ∗ )q ∗ − τ a (0))
+ (V e|bb − V e|ba + b∆V aa − a∆V ab )(1 − τ ∗ )q ∗
+ (1 − τ ∗ )βq ∗ (2q ∗ − 1) ∆V aa − ∆V ab .
The function Λ is linear in τ ∗ . We have
Λ(1) = (a∆V aa − b∆V ab )(1 − τ a (0)) > 0,
therefore, if Λ(0) > 0 then we deduce Λ(τ ∗ ) > 0 ∀τ ∗ . Let us perform Λ(0),
Λ(0) = (a∆V aa − b∆V ab )(q ∗ − τ a (0)) + (V e|bb − V e|ba + b∆V aa − a∆V ab )q ∗
+ βq ∗ (2q ∗ − 1) ∆V e|aa − ∆V e|ab .
= 2βq ∗ 2 (∆V aa − ∆V ab ) + q ∗ (∆V aa − ∆V ab )(a + b − β) + V e|bb − V e|ba
− τ ∗a (0)(a∆V aa − b∆V ab ).
This is a polynomial function of q ∗ . It is convex, negative at q ∗ = 0 and for q ∗ = 1
Λ(0)|q∗ =1 = (1 − τ ∗a (0))(a∆V e|aa − b∆V e|ab ) + β(∆V e|aa − ∆V e|ab )
+ b∆V aa − a∆V ab + V e|bb − V e|ba .
which is positive given that τ ∗a (0) < 1 and that τ ∗b (1) > 0. This polynomial is positive if and only
if q ∗ > q̃, with
√
(∆V aa − ∆V ab )(β − (a + b)) − V e|bb − V e|ba + D0
q̃ =
,
4β(∆V e|aa − ∆V e|ab )
2
with D0 = (∆V aa − ∆V ab )(a + b − β) + (V e|bb − V e|ba )
+ 8βτ ∗a (0)(∆V e|aa − ∆V e|ab )(a∆V e|aa − b∆V e|ab ).
Then,
q ∗ > q̃
41
is equivalent to
√
D>
√ 0
D
which is true given that τ ∗a (0) < 1.
4. Conclusion.
Now we conclude that, when parameters are such that C1 and C2 are satisfied and that
0
(i) γ < min{γ̃, γ },
e
(ii) θ > θ,
urban equilibria (q1 , q2 ) = (q ∗ , q ∗ ) and (q1 , q2 ) = (0, q ∗ ) are both stable steady states.
An example of some parameters combination for which these conditions hold is ∆V aa = 0.07,
∆V ab = 0.01, V e|bb − V e|ba = 0.0005, a = 0.005, b = 0.004, β = 0.008, θ = 0.05, γ = 0.37,
r = 30, α = 1.1.
7.8
Proof of Proposition 5
1. Suppose that Q0 ∈]q̂, 2q̂[.
(i) If segregation holds the urban equilibrium is such that q 1 = Q ∈]q̂, 2q̂[ and q 2 = 0. Given
the function Ψ(.) in proof of Proposition 2, the dynamics of cultural traits is described by
Qt+1 = Ψ(Qt ) + Ψ(0) = Ψ(Qt ).
From proof of Proposition 2, we have that q ∗ is a steady state of the map Ψ and for Q0 ∈]q̂, q ∗ [ the
sequence Qt with dynamics captured by the map Ψ is increasing, while for Q0 ∈]q ∗ , 1[, the sequence
Qt is decreasing. By the continuity of the map Ψ, we deduce that for any Q0 ∈]q̂, 1[, the sequence
Qt converges to Q = q ∗ . Hence for any Q0 ∈]q̂, 2q̂[, the sequence Qt converges to Q = q ∗ .
(ii) If integration holds the urban equilibrium is such that q 1 = q 2 = Q/2 < q̂. The dynamics
42
of cultural traits is described by
Qt+1 = 2Ψ
Qt
2
.
From proof of Proposition 2, we know that 0 is a steady state of the map Ψ. Also, we can deduce
that for Q0 < 2q̂ or equivalently Q0 /2 < q̂, the sequence Qt is decreasing. By the continuity of the
map Ψ, we deduce that for any Q0 ∈]q̂, 2q̂[, the sequence Qt converges to Q = 0.
We conclude that the steady state size of group a under segregation q ∗ is higher than the size
reached under integration which is zero. Since the long-run level of education is an increasing function of Q, we deduce that segregation maximizes the long-run level of education.
2. Suppose that Q0 ∈]2q̂, 1[.
(i) If segregation holds the urban equilibrium is such that q 1 = Q ∈]2q̂, 1[ and q 2 = 0. The dynamics of cultural traits is described by
Qt+1 = Ψ(Qt ).
Following point 1.(i) we deduce that the sequence Qt converges to Q = q ∗ .
(ii) If integration holds the urban equilibrium is such that q 1 = q 2 = Q/2 > q̂. The dynamics
of cultural traits is described by
Qt+1 = 2Ψ
Qt
2
.
From proof of Proposition 2, we know that q ∗ is a steady state of the map Ψ. Following point 1.(i)
and 1.(ii) we deduce that the sequence Qt converges to Q = 2q ∗ .
We conclude that the steady state size of group a under integration 2q ∗ is higher than the size
reached under segregation q ∗ . Since the long-run level of education is an increasing function of Q,
we deduce that segregation maximizes the long-run level of education.
43

Download Report

Cultural Dynamics, Social Mobility and Urban

Paperzz.com

Your Paperzz